The ratio of angular speeds of minute hand and hour hand of a watch is

Options

A. `1:12`
B. `6:1`
C. `12:1`
D. `1:6`

Explanation

To find the ratio of the angular speeds of the minute hand and the hour hand of a watch, we can follow these steps:

### Step 1: Understand Angular Speed
Angular speed (ω) is defined as the angle rotated per unit time. For a complete rotation, the angle is \(2\pi\) radians.

### Step 2: Calculate Angular Speed of the Minute Hand
The minute hand completes one full rotation (i.e., \(2\pi\) radians) in 60 minutes. Therefore, the angular speed of the minute hand (\(ω_{minute}\)) can be calculated as:
\[ ω_{minute} = \frac{2\pi \text{ radians}}{60 \text{ minutes}} = \frac{2\pi}{60} \text{ radians/minute} \]

### Step 3: Calculate Angular Speed of the Hour Hand
The hour hand completes one full rotation (i.e., \(2\pi\) radians) in 12 hours. Since there are 60 minutes in an hour, the total time in minutes for one complete rotation of the hour hand is \(12 \times 60 = 720\) minutes. Therefore, the angular speed of the hour hand (\(ω_{hour}\)) can be calculated as:
\[ ω_{hour} = \frac{2\pi \text{ radians}}{720 \text{ minutes}} = \frac{2\pi}{720} \text{ radians/minute} \]

### Step 4: Find the Ratio of Angular Speeds
Now, we can find the ratio of the angular speeds of the minute hand to the hour hand:
\[ \text{Ratio} = \frac{ω_{minute}}{ω_{hour}} = \frac{\frac{2\pi}{60}}{\frac{2\pi}{720}} \]

### Step 5: Simplify the Ratio
When we simplify the ratio, the \(2\pi\) cancels out:
\[ \text{Ratio} = \frac{1/60}{1/720} = \frac{720}{60} = 12 \]

### Final Answer
Thus, the ratio of the angular speeds of the minute hand to the hour hand is:
\[ \text{Ratio} = 12:1 \]

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